Analysis of Beam Deflection Measurements in the Presence of Linear Absorption

We develop a series of analytical approximations allowing for rapid extraction of the nonlinear parameters from beam deflection measurements. We then apply these approximations to the analysis of cadmium silicon phosphide and compare the results against previously published parameter extraction methods and find good agreement for typical experimental conditions. © 2017 Optical Society of America OCIS codes: (190.4400) Nonlinear optics, materials; (190.3270) Kerr effect. References and links 1. D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010). 2. C. B. de Araújo, A. S. L. Gomes, and G. Boudebs, “Techniques for nonlinear optical characterization of materials: a review,” Rep. Prog. Phys. 79(3), 036401 (2016). 3. J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013). 4. B. A. Ruzicka, S. Wang, J. Liu, K.-P. Loh, J. Z. Wu, and H. Zhao, “Spatially resolved pump-probe study of single-layer graphene produced by chemical vapor deposition [Invited],” Opt. Mater. Express 2(6), 708–716 (2012). 5. E. Dremetsika, B. Dlubak, S. P. Gorza, C. Ciret, M. B. Martin, S. Hofmann, P. Seneor, D. Dolfi, S. Massar, P. Emplit, and P. Kockaert, “Measuring the nonlinear refractive index of graphene using the optical Kerr effect method,” Opt. Lett. 41(14), 3281–3284 (2016). 6. D. S. Kummli, H. M. Frey, and S. Leutwyler, “Femtosecond degenerate four-wave mixing of carbon disulfide: high-accuracy rotational constants,” J. Chem. Phys. 124(14), 144307 (2006). 7. M. R. Ferdinandus, H. Hu, M. Reichert, D. J. Hagan, and E. W. Van Stryland, “Beam deflection measurement of time and polarization resolved ultrafast nonlinear refraction,” Opt. Lett. 38(18), 3518–3521 (2013). 8. R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” Quantum Electronics, IEEE Journal of 38(9), 1205–1216 (2002). 9. M. Reichert, H. Hu, M. R. Ferdinandus, M. Seidel, P. Zhao, T. R. Ensley, D. Peceli, J. M. Reed, D. A. Fishman, S. Webster, D. J. Hagan, and E. W. Van Stryland, “Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide,” Optica 1(6), 436 (2014). 10. G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013). 11. G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015). 12. K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016). 13. M. Reichert, P. Zhao, J. M. Reed, T. R. Ensley, D. J. Hagan, and E. W. Van Stryland, “Beam deflection measurement of bound-electronic and rotational nonlinear refraction in molecular gases,” Opt. Express 23(17), 22224–22237 (2015). 14. M. R. Ferdinandus, H. Hu, M. Reichert, Z. Wang, D. J. Hagan, and E. Van Stryland, “Beam deflection measurement of time and polarization resolved nonlinear refraction,” in Frontiers in Optics 2013, I. R. D. A. N. Kang, and D. Hagan, eds. (Optical Society of America, Orlando, Florida, 2013), p. FTh4C.2. Vol. 7, No. 5 | 1 May 2017 | OPTICAL MATERIALS EXPRESS 1598 #286316 https://doi.org/10.1364/OME.7.001598 Journal © 2017 Received 7 Feb 2017; revised 30 Mar 2017; accepted 30 Mar 2017; published 12 Apr 2017 15. N. Kinsey, A. A. Syed, D. Courtwright, C. DeVault, C. E. Bonner, V. I. Gavrilenko, V. M. Shalaev, D. J. Hagan, E. W. Van Stryland, and A. Boltasseva, “Effective third-order nonlinearities in metallic refractory titanium nitride thin films,” Opt. Mater. Express 5(11), 2395 (2015). 16. S. C. Kumar, M. Jelínek, M. Baudisch, K. T. Zawilski, P. G. Schunemann, V. Kubeček, J. Biegert, and M. Ebrahim-Zadeh, “Tunable, high-energy, mid-infrared, picosecond optical parametric generator based on CdSiP2,” Opt. Express 20(14), 15703–15709 (2012). 17. V. Kemlin, B. Boulanger, V. Petrov, P. Segonds, B. Ménaert, P. G. Schunneman, and K. T. Zawilski, “Nonlinear, dispersive, and phase-matching properties of the new chalcopyrite CdSiP2 [Invited],” Opt. Mater. Express 1(7), 1292–1300 (2011). 18. K. T. Zawilski, P. G. Schunemann, T. C. Pollak, D. E. Zelmon, N. C. Fernelius, and F. Kenneth Hopkins, “Growth and characterization of large CdSiP2 single crystals,” J. Cryst. Growth 312(8), 1127–1132 (2010). 19. K. Kato, N. Umemura, and V. Petrov, “Sellmeier and thermo-optic dispersion formulas for CdSiP2,” J. Appl. Phys. 109(11), 116104 (2011). 20. G. Ghosh, “Sellmeier coefficients for the birefringence and refractive indices of ZnGeP2 nonlinear crystal at different temperatures,” Appl. Opt. 37(7), 1205–1212 (1998). 21. S. Chaitanya Kumar, J. Krauth, A. Steinmann, K. T. Zawilski, P. G. Schunemann, H. Giessen, and M. EbrahimZadeh, “High-power femtosecond mid-infrared optical parametric oscillator at 7 μm based on CdSiP2,” Opt. Lett. 40(7), 1398–1401 (2015). 22. S. C. Kumar, M. Jelínek, M. Baudisch, K. T. Zawilski, P. G. Schunemann, V. Kubeček, J. Biegert, and M. Ebrahim-Zadeh, “Tunable, high-energy, mid-infrared, picosecond optical parametric generator based on CdSiP2,” Opt. Express 20(14), 15703–15709 (2012).


Introduction
Knowledge of the temporal response of the Nonlinear Refraction (NLR) and Nonlinear Absorption (NLA) of materials is key for understanding of the physical mechanisms underlying the Nonlinear Optical (NLO) properties [1]. Various experimental techniques have been developed for measuring this response [2]. Commonly used methods such as pumpprobe [3,4] provide the temporal response of the NLA, the refraction via the induced birefringence as in the Optical Kerr Effect (OKE) experiment [5] or with use of a local oscillator as with four wave mixing [6]. The Beam Deflection (BD) method was developed as a high sensitivity, easy to implement time and polarization resolved technique for simultaneous measurement of NLR and NLA [7].
Previously, techniques for analyzing BD data for instantaneous and non-instantaneous nonlinearities in the presence of Group Velocity Mismatch (GVM) have been determined [8,9]. This method, however is limited in that it only treats materials in the undepleted excitation approximation, where the absorption is sufficiently small so that irradiance throughout the sample is constant. This is a significant limitation, especially in probing 2-D, plasmonic and metamaterials which typically have high linear and nonlinear absorption [10,11]. These materials have come to be of interest to the NLO community for a variety of application such as photonic-electronic interconnects, all optical switching and computing and hybrid silicon photonics [12].
Additionally, depending on the spatial and temporal resolution required, this method can be very time consuming and makes the extraction of NLO parameters tedious, especially for materials with multiple mechanisms acting together. In this work we develop an analytic approach that accounts for change in the excitation throughout the sample due to linear, absorption. To demonstrate the validity of these approximations, we will compare them against previous extraction methods to show good agreement for typical experimental conditions. We then apply our approximations to measurements of Cadmium Silicon Phosphide (CSP), a material with large index dispersion and linear absorption.

Analysis of beam deflection data
The BD method operates by using a strong Gaussian excitation beam to generate an index change ∆n within the material as seen in Fig. 1. If the spot size of the probe is much smaller than the excitation, the probe, displaced from the peak by half the beam waist of the excitation, experiences a prism-like index gradient. This transient prism deflects the beam by some angle ∆θ. This deflection angle is in turn in measured by the quadrant cell diode as a change in the differential energy signal of the probe ∆E p [τ d ] = E p,L -E p,R, where E p,(L,R) are the probe signal measured from the left and right sides of the quadrant diode and τ d is the delay between the excitation and probe. This signal is normalized by the total energy of the probe E p [τ d ] = E p,L + E p,R so that the signal ∆E p / E p [τ d ] is proportional to ∆n [13]. Similarly, the NLA can be extracted using total energy signal E p [τ d ] which can be normalized to calculate the transmission. Following the method outlined in Reichert et al [9], we start with the propagation equation in the absence of Group Velocity Dispersion (GVD) in normalized coordinates.
where a p [r,Z,t] is the dimensionless field of the probe, Z = z / L is the normalized propagation distance, L is the sample length, τ = t / τ e is the normalized temporal coordinate, τ e is the excitation pulse duration σ p = α p L / 2 is the normalized linear absorption of the probe, ρ = Δn g L / (τ e c) is the GVM parameter, Δn g is the difference in the group indices of the excitation and probe, c is the speed of light in vacuum and α p is the linear absorption of the probe.
where a e [r,Z,τ] is the field of the excitation, shifted so the probe is centered over the maximum gradient of the spatial envelope (X 0 = ½), a p [r,0,τ] is the field of the probe at the front of the sample, and X = x / w e , Y = y / w e are the normalized spatial coordinates, σ e = α e L / 2 is the normalized linear absorption parameter of the excitation, α e is the linear absorption of the excitation, T = τ p / τ e is the ratio of the pulse durations, W = w 0,p / w e is the ratio of the spot sizes of the beams, and w 0,p and w e are the 1 / e 2 spot sizes of the probe and excitation. We can solve for the field at the back of the sample a p [r,1,τ] in the linear (L << z 0,p ) and nonlinear (L << z 0,p / Δφ 0 ) thin sample approximations where z 0,p is the Rayleigh range of the probe and Δφ 0 is the peak induced phase shift. Typically, Δφ 0 << 1 and L can be selected to meet these conditions giving us where H[τ] is the integral for the pulse overlap in the sample accounting for GVM and depletion of the excitation due to linear absorption. The real part H´[τ] corresponds to the change in the absorption Δα p and H´´[τ] corresponds to the nonlinear phase accumulation Δφ p . As in previous work the signal can be calculated by Fresnel propagating a p [r,1,τ] and spatially integrating over both sides of the detector and temporally over the pulse duration. In this analysis we use the convenience that the Fresnel propagation to the detector will yield another Gaussian expanded in size, deflected by an angle ∆θ due to H´´[τ] and attenuated due to H´ [τ]. The As shown in Fig. 2, the effect of this attenuation is to make it appear as if the probe beam has been translated slightly, without distorting its shape significantly. Thus while φ[r,τ] acts as the prism that deflects the probe an angle ∆θ, Q[r,τ] attenuates and slightly translates the probe yielding a small deflection signal due to NLA.
In the case of no linear absorption (σ e = 0, F → 1), excitation much larger than the probe (W << 1, R → 1), and negligible NLA (Γ ≈0), Eq. (11) reduces to the previously derived expression for the signal in transparent material [14]. Note that by setting X 0 = 0, Eq. (10) can be used to calculate the expression for the transmission in excite -probe experiments [8] [ ] Taking the ratio of the deflection signal due to the refraction (ΔE p / E p [Γ = 0]) and absorption As we seen in Fig. 2 the effect of a small translation is reduced by the expansion of the probe as it propagates to the detector, while the translation due to the deflection scales with distance. For a typical configuration (D > 15) we can effectively set H´[τ] = 0 in Eq. (9). This makes the BD method particularly well suited for measuring materials with large NLA since the refraction and absorption signals are essentially independent of each other. This approach has been applied to BD measurements of thin film refractory metal nitrides, which possess both high linear and nonlinear absorption [15].

Comparisons to previous models
In Fig. 3 we compare the method from Reichert et al and our analysis for various values of ρ, along with our approximations compared to the Fresnel propagation method. The agreement between the Fresnel propagation and the analytical expression is very good, with a difference at the peak signal of less than 2.0%. As seen in Fig. 4, the effect of GVM is to extend the temporal range of the signal, as one would expect as it is possible for the excitation pulse to walk entirely through the probe pulse over a wide range of delays [8]. The effect of the excitation depletion is to reduce the signal at negative delays due to the excitation catching up to the probe at the back of the sample after it has been significantly attenuated.

Measurements of cadmium silicon phosphide
To test the validity of our analysis, we fit measurements of CSP using both the above expressions. CSP is a II-IV-V 2 chalcopyrite semiconductor with a high 2nd order nonlinear coefficient (d 36 ) of 84.5 pm/V in the mid-infrared spectrum [16] with sufficient birefringence for 2 µm to mid-IR wavelength conversion [17]. Grown by the horizontal gradient freeze technique, CSP shows significantly improved transparency (i.e., lower optical absorption), unfortunately combined with a somewhat lower thermal conductivity, as compared to the more established Zinc Germanium Phosphide (ZGP) [18]. These properties suggest that CSP will display a higher thermal-lensing threshold and thus enable higher power mid-IR laser output than possible with ZGP [19,20]. Previous works have demonstrated CSP based high power femtosecond Optical Parametric Amplifiers (OPA) pumped at 1064 nm [21].
BD measurements are made using a Ti:Sapphire amplified system (KM Labs Wyvern 1000-10) producing 4.2 mJ, 35 fs (FWHM) pulses at 790 nm operating at a 1 kHz repetition rate. The strong excitation pulse is obtained by splitting off ~5 μJ of the fundamental with a beam splitter. An optical parametric generator/amplifier (Light Conversion TOPAS-Prime) is pumped by the fundamental to generate the 650 nm, 55 fs probe pulses, which is then spatially filtered to produce a Gaussian irradiance profile. The probe is focused to a spot size, w p ~3 -5 times smaller than w e , both which were determined by knife-edge scans. The probe is displaced from the peak of the excitation by Δx = ½w e to the maximized irradiance gradient where the probe experiences an induced refractive index gradient, causing it to be deflected by a small angle ∆θ. The deflection signal is measured using a quad-segmented Si photodiode (OSI QD50-0-SD) which simultaneously measures ΔE p [τ d ] and E p [τ d ], each of which is detected via lock-in detection (Stanford Systems SR-830). A mechanical optical chopper (Thorlabs MC-2000) synchronized with the excitation repetition rate is used to modulate the excitation at 286 Hz. Fig. 5. Fit of CSP data using analytic approximation for a) transmission and b) deflection. Note that the reduction in the signal between τ d = 5 and τ d = 25 is due to the depletion of the excitation. The difference in the slope of the rise and fall of the signal is due to GVD, which is not accounted for. Inset) absorption coefficient α e vs. peak excitation irradiance I 0,e . Plots have been shifted vertically by 2.5% for clarity.  [22]. The deviation of the fit from the data at negative delay is due to the GVD, which broadens the pulses as they propagate through the material, thus reducing the irradiance and the induced nonlinear effect. This effect is not modeled in order to derive an analytic solution for Eq. (1). It may be possible to determine the GVD either through ellipsometry measurements or by BD measurements of differing thicknesses of material.

Conclusions
We have extended the analysis of BD data to include depletion of the excitation due to linear absorption. Additionally, we have applied a series of approximations in order to arrive at analytic expressions that are much quicker to evaluate than the standard Fresnel propagation based methods. We have shown that the approximations have very good agreement with the Fresnel propagation method, with a difference of less than 2%. Lastly we have applied these expressions to the extraction of the nonlinear properties of CSP.